Average slope

Start with:

Nearby point:

Average slope:

Average slope

Start with a point \((x,f(x))\) on the graph.

A nearby point can be \((x+\Delta x, f(x+\Delta x))\), where \(\Delta x\) is a short distance from \(x\).

The average slope between \((x,f(x))\) and \((x+\Delta x, f(x+\Delta x))\) is given by

\[\text{average slope} = \frac{f(x+\Delta x) - f(x)}{(x+\Delta x)-x} = \frac{f(x+\Delta x) - f(x)}{\Delta x}.\]

What happens when \(\Delta x\) becomes smaller and smaller?

What happens when \(\Delta x \to 0\)

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